Monte Carlo Visulization

Monte Carlo simulations are a class of computational algorithms that rely on repeated random sampling to obtain numerical results. They are particularly useful for understanding the behavior of systems too complex for analytical solutions. In this blog post, we’ll explore how to visualize Monte Carlo simulations using R, with the help of the gganimate, tidyverse, gridExtra, and magick packages.

Several notes:

When using gganimate, the gifski renderer is not consistent. It may miss frames from to time. It appears the most consistent way of generating animation is - Use transistion_manual - Use file_renderer to generate PNG files - Review PNG files, edit as needed. - Generate gif with magick package. ## Setting the Scene To ensure reproducibility, we set a seed for R’s random number generator. We also define the precision of our output for clarity and consistency.

Case Study - Tolerance Analysis.

In this example, we use a statistical tolerance analysis of the voltage divider as an example.

The voltage division ratio \(G = \frac{R1}{R1 + R2}\), where \(R_1\) and \(R_2\) are independent variables following normal distributions. Our goal is to simulate this process a million times and visualize the distribution of G.

N_SIM <- 1e6
mcHist <- tibble(R1 = rnorm(N_SIM, 1, 0.04/12),
                R2 = rnorm(N_SIM, 1, 0.04/12)) %>%
  mutate(G = R1 /(R1+R2)) %>% 
  mutate(index = row_number()) %>% 
  pivot_longer(cols = c("R1", "R2", "G"),
               names_to = "Variable", 
               values_to = "value") %>% 
  mutate(round = if_else(Variable == "G", 
                             plyr::round_any(value, 5e-4),
                             plyr::round_any(value, 1e-3)))

Create Frequency Table

count_by_round <-mcHist %>%
  select(Variable, round) %>%
  group_by(Variable, round) %>%
  summarise(n = n(), .groups = "drop") %>%
  ungroup()

gplot <- count_by_round %>%
  filter(Variable == "G") %>% 
  ggplot(aes(round, n)) +
  facet_grid(~Variable) +
  geom_bar(stat = "identity", color="black") + 
  geom_vline(xintercept = 0.49, color="blue") + 
  geom_vline(xintercept = 0.51, color="blue") +
  theme_bw() + 
  theme(strip.text = element_text(size = 16, hjust = 0.1),
        axis.text.y = element_blank(),
        axis.text.x = element_text(angle=90, vjust=.5, hjust=1))

rplot <- count_by_round %>%
  filter(Variable != "G") %>% 
  ggplot(aes(round, n)) +
  facet_grid(~Variable) +
  geom_bar(stat = "identity", color="black") + 
  geom_vline(xintercept = 0.98, color="blue") + 
  geom_vline(xintercept = 1.02, color="blue") +
  theme_bw() + 
  theme(strip.text = element_text(size = 16, hjust = 0.1),
        axis.text.y = element_blank(),
        axis.text.x = element_text(angle=90, vjust=.5, hjust=1))

Combine Plots

# Combine plots
plot = grid.arrange(rplot, gplot, nrow = 2)

# alternatively, use geom_bar to generate histogram from raw data
gg <- mcHist %>%
  filter(Variable == "G") %>%
  ggplot(aes(x=round)) + 
  geom_bar(color="black", stat = "count", alpha = .3, fill = "transparent") +
  labs(x = "", y = "Frequency") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        strip.text = element_text(size = 16, hjust = 0.1),
        axis.text.y = element_blank()) +
  geom_vline(xintercept = 0.49, color="blue") + geom_vline(xintercept = 0.51, color="blue")

gg

Animating the Simulation

Animations can provide a dynamic view of the simulation process. Here, we’ll create several animations to illustrate different aspects of the Monte Carlo simulation. Preparation Slice the simulated data into (N_Frames — 1) slices, where the first frame has zero simulations.

# constants for animation
FPS = 10
N_FRAME = 201

count_by_index <- function(mcHist, nframes=120) {
  # Initialization of variables
  data <- tibble(idx = integer(),
                 n_sample = integer(),
                 Variable = character(),
                 round = numeric(),
                 n = integer())
  N_SIM = max(mcHist$index)
  step <- floor(N_SIM / (nframes-1))
  data <- data %>% 
    add_row(tibble(idx = 1,
                 n_sample = 0,
                 Variable = c("G","R1", "R2"),
                 round = c(0.5, 1.0, 1.0),
                 n = 0))

  for (i in 1:nframes-1) {
    # Data manipulation and summarization steps
    row <- mcHist %>%
           select(index, Variable, round) %>%
           filter(index <= step * i) %>%
           group_by(Variable, round) %>%
           summarise(n = n(), .groups = 'drop') %>%
           mutate(idx = i + 1,
                  n_sample = floor(step * i))
    # Append the summary data to the main data frame
    data <- data %>%
            add_row(row)
  }
  return(data)
}

hist_anim_data <- count_by_index(mcHist, N_FRAME) %>% 
  arrange(idx)

Create Text for G and R

In this section, we will create a text gif to demonstrate the concept of a transfer function, which takes values of x to generate corresponding values of y.

textplot_data <- mcHist %>% 
  filter(index<=5) %>% 
  arrange(index, Variable)
# dot plot for first 5 samples 
tgplot <- textplot_data %>%
  filter(Variable == "G") %>%
  ggplot(aes(group = index, x = round)) + # group by index is important
  facet_grid(~Variable) +
  geom_point(aes(y=0), size = 5, color="black") + 
  geom_text(aes(x=0.5, y=0.1, label = round), 
                position = position_dodge(width=0.9), color="blue", size = 10) +
  labs(x = "", y = "Frequency") +
  xlim(0.49, 0.51) +
  theme_bw() +
  theme(panel.grid = element_blank(),
    strip.text = element_text(size = 16, hjust = 0.1),
    axis.text.y = element_blank()) 

trplot <- textplot_data %>%
  filter(Variable != "G") %>%
  ggplot(aes(group = index, x = round)) + # group by index is important
  facet_grid(~Variable) +
  geom_point(aes(y=0), size = 5, color="black") + 
  geom_text(aes(x=1, y=0.1, label = round), 
                position = position_dodge(width=0.9), color="blue", size = 10) +
  labs(x = "", y = "Frequency") +
    xlim(0.98, 1.02) +
    theme_bw() +
    theme(
      panel.grid = element_blank(),
      strip.text = element_text(size = 16, hjust = 0.1),
      axis.text.y = element_blank()
    )

Create GIF

Use transition_manual(frames = index) all through to avoid frames misalignment when using transition_states(states = Time)

# Create GIF
r_anim <- trplot + transition_manual(frames = index) 
rgif <- animate(r_anim,
                width = 800, height = 300,
                nframes = 5, 
                fps = 2.5,
                #renderer = magick_renderer()
                renderer = gifski_renderer("./anim/tr.gif")
                # renderer = file_renderer("./frames", prefix = "tr")
                )

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g_anim <- tgplot + transition_manual(frames = index)
ggif <- animate(g_anim,
                width = 800, height = 300,
                nframes = 5,
                fps = 2.5,
                renderer = gifski_renderer("./anim/tg.gif")
                # renderer = magick_renderer()
                # renderer = file_renderer("./frames", prefix = "tg")
                )

Combine 2 gifs

stack2gif <- function(gif1, gif2, n_frames){
  # read gif as imageck format 
  a_mgif <- image_read(gif1) 
  b_mgif <- image_read(gif2)
  new_gif <- image_append(c(a_mgif[1], b_mgif[1]), stack = TRUE)
  for(i in 2:n_frames) {
    combined <- image_append(c(a_mgif[i], b_mgif[i]), stack = TRUE)
    new_gif <- c(new_gif, combined)
    }
return(new_gif)  

a_mgif <- image_read(rgif)
b_mgif <- image_read(ggif)
new_gif <- image_append(c(a_mgif[1], b_mgif[1]), stack = TRUE)
for(i in 2:5) {
  combined <- image_append(c(a_mgif[i], b_mgif[i]), stack = TRUE)
  new_gif <- c(new_gif, combined)
}
anim_save("./anim/5dots.gif", new_gif)

}

The created gif:

Create dot histogram

In this section, we create a histogram gif, where each dot represent one data point.

first_n = 100

count_data_n <- mcHist %>% 
  filter(index <= first_n) %>% 
  arrange(index, Variable) %>% 
  group_by(Variable, round) %>% 
  mutate(y = seq_along(round))

fg_plot <- count_data_n %>%
  filter(Variable == "G") %>%
  ggplot(aes(group = index, round, y)) + # group by index is important
  facet_grid(~Variable) +
  geom_point(size = 2) +
  labs(x = "", y = "Frequency") +
  theme_bw() +
  theme(
    panel.grid = element_blank(),
    strip.text = element_text(size = 32, hjust = 0.1),
    axis.text.y = element_blank()
  )

plot2frames <- function(plot, 
                     prefix, 
                     w = 800, h = 300,
                     path_frames = "./frames"
                     ){
  anim <- plot + transition_manual(index, cumulative = TRUE)
  gif <- animate(
    anim,
    width = w,
    height = h,
    renderer = file_renderer(path_frames, prefix = prefix)
  )
  return(gif)
} 


fg_anim <- plot2frames(fg_plot, prefix = "gdot")

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fr_plot <- count_data_n %>%
  filter(Variable != "G") %>%
  ggplot(aes(group = index, round, y)) + # group by index is important
  facet_grid(~Variable) +
  geom_point(size = 2) +
  labs(x = "", y = "Frequency") +
  theme_bw() +
  theme(
    panel.grid = element_blank(),
    strip.text = element_text(size = 32, hjust = 0.1),
    axis.text.y = element_blank()
  )


fg_anim <- plot2frames(fg_plot, prefix = "gdot")

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fr_anim <- plot2frames(fr_plot, prefix = "rdot")

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fn_gif <- stack2gif(fr_anim, fg_anim, first_n) %>% 
  image_animate(
    fps = FPS,
    loop = 0,
  )

anim_save("./anim/fn.gif", fn_gif)

This gif shows how first N simulation accumated to form the historgram.

Historgram bar gif

gplot <-
  hist_anim_data %>%
  filter(Variable == "G") %>%
  ggplot(aes(group = idx, round, n)) + # group by index is important
  facet_grid(~Variable) +
  geom_bar(stat = "identity") +
  geom_vline(xintercept = 0.49, color="blue") +
  geom_vline(xintercept = 0.51, color="blue") +
  labs(x = "", y = "Frequency") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        strip.text = element_text(size = 32, hjust = 0.1),
        plot.title = element_text(size = 32, hjust = 0.1),
        axis.text.y = element_blank())

step <- N_SIM / (N_FRAME - 1)
gbar_anim <- gplot + 
    ggtitle('{format(floor((frame - 1) * step), scientific = FALSE, big.mark = "," )} Samples Generated') + 
  transition_manual(frames = idx)
gbar_gif <- animate(gbar_anim,
                    nframes = N_FRAME,
                    width = 800, height = 300,
                    # fps = FPS,
                    # renderer = gifski_renderer("./anim/gbar.gif"),
                    # renderer = magick_renderer()
                    renderer = file_renderer("./frames", prefix = "gbar")
)

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# anim_save("./anim/gbar.gif", gbar_gif)
### For R1 and R2
rplot <-
  hist_anim_data %>%
  filter(Variable != "G") %>%
  ggplot(aes(group = idx, round, n)) + # group by index is important
  facet_grid(~Variable) +
  geom_bar(stat = "identity") +
  geom_vline(xintercept = 0.98, color="blue") +
  geom_vline(xintercept = 1.02, color="blue") +
  labs(x = "", y = "Frequency") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        strip.text = element_text(size = 32, hjust = 0.1),
        axis.text.y = element_blank())

rbar_anim <- rplot + transition_manual(frames = idx)
rbar_gif <- animate(rbar_anim,
                    nframes = N_FRAME,
                    width = 800, height = 300,
                    # fps = FPS,
                    # renderer = gifski_renderer("./anim/rbar.gif")
                    # renderer = magick_renderer()
                    renderer = file_renderer("./frames", prefix = "rbar")
                    )

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Warning: file_renderer failed to copy frames to the destination directory

Combine 2 gifs

a_mgif <- image_read(rbar_gif)
b_mgif <- image_read(gbar_gif)
# c_mgif <- image_read(counter_gif)

new_gif <- image_append(c(a_mgif[1], b_mgif[1]), stack = TRUE)
for(i in 2:N_FRAME) {
  combined <- image_append(c(a_mgif[i], b_mgif[i]), stack = TRUE)
  new_gif <- c(new_gif, combined)
}
new_gif <- new_gif %>% 
  image_animate(
    fps = FPS,
    loop = 0,
  )

anim_save("./anim/bar.gif", new_gif)

We check the generated gif:

Reference

• Roh’s Animation, without using gganimate
• Stackoverflow - Animated histogram, using `gganimate